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arXiv · 2310.01817

The Note on the Closure of Continuous Functions in Variable-Exponent Lebesgue Spaces for Multiple Variables

Nikoloz Devdariani
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Abstract

In this paper, we generalize a recently obtained result by Kopaliani and Zviadadze from the one-variable case to the several-variable case. Specifically, in terms of decreasing rearrangement, we characterize those exponents $p(\cdot)$ for which the corresponding variable-exponent Lebesgue space $L^{p(\cdot)}([0;1]^n)$ shares the property with $L^\infty([0;1]^n)$ such that the space of continuous functions $C([0;1]^n)$ forms a closed linear subspace in $L^{p(\cdot)}([0;1]^n)$ . In particular, we derive the necessary and sufficient conditions on the decreasing rearrangement of the exponent $p(\cdot)$ for which there exists an equimeasurable exponent of $p(\cdot)$ such that the corresponding variable-exponent Lebesgue space possesses the aforementioned property.

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Nikoloz Devdariani. 2023-10-03. The Note on the Closure of Continuous Functions in Variable-Exponent Lebesgue Spaces for Multiple Variables. https://arxiv.org/abs/2310.01817

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